IsotropicEntanglement - Fields Institute · Gap of Ignorance Philosophy of Isotropic Entanglement...
Transcript of IsotropicEntanglement - Fields Institute · Gap of Ignorance Philosophy of Isotropic Entanglement...
Gap of IgnorancePhilosophy of Isotropic Entanglement
Theory of Isotropic EntanglementThe Method of Isotropic Entanglement
More Numerical results
Isotropic Entanglement(Density of States of Quantum Spin Systems)
Ramis Movassagh1 and Alan Edelman2
1Department of Mathematics, Northeastern University2Department of Mathematics, M.I.T.
Fields Institute, July 2013
Ramis Movassagh arXiv: 1012.5039; Phys. Rev. Lett. 107, 097205 (2011)
Gap of IgnorancePhilosophy of Isotropic Entanglement
Theory of Isotropic EntanglementThe Method of Isotropic Entanglement
More Numerical results
The eigenvalue distribution: Motivation
Synonymous: (Energy) Spectrum; eigenvalue distribution, densityof states, level densities etc.
First step for all eigenvalue problems (e.g. quantummechanics) of sums of matricesPhysical: Partition function and therefore the thermodynamicsof QMBS
Ramis Movassagh arXiv: 1012.5039; Phys. Rev. Lett. 107, 097205 (2011)
Gap of IgnorancePhilosophy of Isotropic Entanglement
Theory of Isotropic EntanglementThe Method of Isotropic Entanglement
More Numerical results
The eigenvalue distribution: Motivation
Synonymous: (Energy) Spectrum; eigenvalue distribution, densityof states, level densities etc.
First step for all eigenvalue problems (e.g. quantummechanics) of sums of matricesPhysical: Partition function and therefore the thermodynamicsof QMBS
Goal: Given the geometry, local spin states, and type of localinteraction, capture the spectrum of the H.
Ramis Movassagh arXiv: 1012.5039; Phys. Rev. Lett. 107, 097205 (2011)
Gap of IgnorancePhilosophy of Isotropic Entanglement
Theory of Isotropic EntanglementThe Method of Isotropic Entanglement
More Numerical results
Complexity issues
Note: Generally the Spectrum of QMBS is hard to find “exactly”(QMA-complete):
F.G.S.L. Brandao’s Thesis (2008).
B. Brown, S. T. Flammia, N. Schuch (2010), “ComputationalDifficulty of Computing the Density of States”.
Phys. Rev. Lett. 107, 040501 (2011)
Ramis Movassagh arXiv: 1012.5039; Phys. Rev. Lett. 107, 097205 (2011)
Gap of IgnorancePhilosophy of Isotropic Entanglement
Theory of Isotropic EntanglementThe Method of Isotropic Entanglement
More Numerical results
Sums of non-commuting Hamiltonians
Figure: Qudit Chain
Hk,k+1 : d2×d2
Generic local terms =⇒ Quantum Spin Glasses.
H =N−1
∑k=1
Idk−1⊗Hk,k+1⊗ IdN−k−1 .
Ramis Movassagh arXiv: 1012.5039; Phys. Rev. Lett. 107, 097205 (2011)
Gap of IgnorancePhilosophy of Isotropic Entanglement
Theory of Isotropic EntanglementThe Method of Isotropic Entanglement
More Numerical results
Non-zero elements of H
0 100 200 300 400 500 600 700 800 900 1000
0
100
200
300
400
500
600
700
800
900
1000
nz = 53248
H = Σk=1
4 H
k,k+1
Ramis Movassagh arXiv: 1012.5039; Phys. Rev. Lett. 107, 097205 (2011)
Gap of IgnorancePhilosophy of Isotropic Entanglement
Theory of Isotropic EntanglementThe Method of Isotropic Entanglement
More Numerical results
Why not just add eigenvalues?
Id×d ⊗Ad×d + Bd×d ⊗ Id×d
Ramis Movassagh arXiv: 1012.5039; Phys. Rev. Lett. 107, 097205 (2011)
Gap of IgnorancePhilosophy of Isotropic Entanglement
Theory of Isotropic EntanglementThe Method of Isotropic Entanglement
More Numerical results
Why not just add eigenvalues?
Id×d ⊗Ad×d + Bd×d ⊗ Id×d
Id×d ⊗Ad2×d2 + Bd2×d2⊗ Id×d
Ramis Movassagh arXiv: 1012.5039; Phys. Rev. Lett. 107, 097205 (2011)
Gap of IgnorancePhilosophy of Isotropic Entanglement
Theory of Isotropic EntanglementThe Method of Isotropic Entanglement
More Numerical results
Interactions: H = ∑N−1k=1
(I⊗Hk ,k+1⊗ I
)= Hodd +Heven
Eigenvectors of odds: QA = Q1⊗Q3⊗·· ·⊗QN−2⊗ Id×d
Qk : d2×d2 matrix of eigenvectors of Hk,k+1
Ramis Movassagh arXiv: 1012.5039; Phys. Rev. Lett. 107, 097205 (2011)
Gap of IgnorancePhilosophy of Isotropic Entanglement
Theory of Isotropic EntanglementThe Method of Isotropic Entanglement
More Numerical results
Interactions: H = ∑N−1k=1
(I⊗Hk ,k+1⊗ I
)= Hodd +Heven
Eigenvectors of odds: QA = Q1⊗Q3⊗·· ·⊗QN−2⊗ Id×d
Eigenvectors of evens: QB = Id×d ⊗Q2⊗Q4⊗·· ·⊗QN−1
Qk : d2×d2 matrix of eigenvectors of Hk,k+1
Ramis Movassagh arXiv: 1012.5039; Phys. Rev. Lett. 107, 097205 (2011)
Gap of IgnorancePhilosophy of Isotropic Entanglement
Theory of Isotropic EntanglementThe Method of Isotropic Entanglement
More Numerical results
H = Hodd +Heven = QAAQ−1A +QBBQ−1B
Change bases such that Hodd is diagonal. Therefore,
��
� H = A+Q−1
q BQq
Qq ≡ (QB)−1QA ∼ N random parameters
Ramis Movassagh arXiv: 1012.5039; Phys. Rev. Lett. 107, 097205 (2011)
Gap of IgnorancePhilosophy of Isotropic Entanglement
Theory of Isotropic EntanglementThe Method of Isotropic Entanglement
More Numerical results
Classical sum
Ramis Movassagh arXiv: 1012.5039; Phys. Rev. Lett. 107, 097205 (2011)
Gap of IgnorancePhilosophy of Isotropic Entanglement
Theory of Isotropic EntanglementThe Method of Isotropic Entanglement
More Numerical results
Isotropic (Free) sum
Ramis Movassagh arXiv: 1012.5039; Phys. Rev. Lett. 107, 097205 (2011)
Gap of IgnorancePhilosophy of Isotropic Entanglement
Theory of Isotropic EntanglementThe Method of Isotropic Entanglement
More Numerical results
Isotropic, Quantum, and Classical
Ramis Movassagh arXiv: 1012.5039; Phys. Rev. Lett. 107, 097205 (2011)
Gap of IgnorancePhilosophy of Isotropic Entanglement
Theory of Isotropic EntanglementThe Method of Isotropic Entanglement
More Numerical results
Quantum as a “sliding” sum of classical and iso
Ramis Movassagh arXiv: 1012.5039; Phys. Rev. Lett. 107, 097205 (2011)
Gap of IgnorancePhilosophy of Isotropic Entanglement
Theory of Isotropic EntanglementThe Method of Isotropic Entanglement
More Numerical results
Local terms: Wishart matrices
0 10 20 30 40 50 600
0.01
0.02
0.03
0.04
0.05N=5, d=2, r=4, trials=500000
Eigenvalue
Density
Classical Approximation
Isotropic Approximation
Isotropic Entanglement (IE)
Ramis Movassagh arXiv: 1012.5039; Phys. Rev. Lett. 107, 097205 (2011)
Gap of IgnorancePhilosophy of Isotropic Entanglement
Theory of Isotropic EntanglementThe Method of Isotropic Entanglement
More Numerical results
The action starts at the fourth moment
Theorem(The Matching Three Moments Theorem) The first threemoments of the quantum, iso and classical sums are equal.
Ramis Movassagh arXiv: 1012.5039; Phys. Rev. Lett. 107, 097205 (2011)
Gap of IgnorancePhilosophy of Isotropic Entanglement
Theory of Isotropic EntanglementThe Method of Isotropic Entanglement
More Numerical results
The Departure Theorem
The Departure Theorem
miso4 = 1
dN E{
Tr[A4+4A3Q−1BQ+4A2Q−1B2Q+4AQ−1B3Q+2(AQ−1BQ)
2+B4
]}
mq4=
1dN E
{Tr[A4+4A3Q−1
q BQq+4A2Q−1q B2Qq+4AQ−1
q B3Qq+2(AQ−1q BQq)
2+B4
]}
mc4=
1dN E{Tr[A4+4A3B+4A2B2+4AB3+2(AB)2+B4]}
Ramis Movassagh arXiv: 1012.5039; Phys. Rev. Lett. 107, 097205 (2011)
Gap of IgnorancePhilosophy of Isotropic Entanglement
Theory of Isotropic EntanglementThe Method of Isotropic Entanglement
More Numerical results
Quantum agony
Ramis Movassagh arXiv: 1012.5039; Phys. Rev. Lett. 107, 097205 (2011)
Gap of IgnorancePhilosophy of Isotropic Entanglement
Theory of Isotropic EntanglementThe Method of Isotropic Entanglement
More Numerical results
Resolving the agony
Lemma: Only these matter
1mETr[(H(3)⊗IdN−2)(I⊗H(4)⊗IdN−3)(H(3)⊗IdN−2)(I⊗H(4)⊗IdN−3)]
= 1d3
{E(H(3)
i3i4,j3j4H(3)i3p4,j3k4
)E(H(4)
j4i5,k4k5H(4)
i4i5,p4k5
)},
Ramis Movassagh arXiv: 1012.5039; Phys. Rev. Lett. 107, 097205 (2011)
Gap of IgnorancePhilosophy of Isotropic Entanglement
Theory of Isotropic EntanglementThe Method of Isotropic Entanglement
More Numerical results
Quantum as a convex combination of classical and iso
Use fourth moments to form a hybrid theory
γq2 = pγ
c2 +(1−p)γ
iso2
γ(•)2 is found from the fourth moments
Ramis Movassagh arXiv: 1012.5039; Phys. Rev. Lett. 107, 097205 (2011)
Gap of IgnorancePhilosophy of Isotropic Entanglement
Theory of Isotropic EntanglementThe Method of Isotropic Entanglement
More Numerical results
The Slider Theorem
Theorem(The Slider Theorem) The quantum kurtosis lies in between theclassical and the iso kurtoses, γ iso
2 ≤ γq2 ≤ γc
2 . Therefore there existsa 0≤ p ≤ 1 such that γ
q2 = pγc
2 +(1−p)γ iso2 . Further,
limN→∞ p = 1.
Ramis Movassagh arXiv: 1012.5039; Phys. Rev. Lett. 107, 097205 (2011)
Gap of IgnorancePhilosophy of Isotropic Entanglement
Theory of Isotropic EntanglementThe Method of Isotropic Entanglement
More Numerical results
Universality of p
Corollary
(Universality) p 7→ p (N,d ,β ), namely, it is independent of thedistribution of the local terms.
101
102
0
0.2
0.4
0.6
0.8
1
N
p
p Vs. N
d=2
d=3
d=4
d=∞
Here β = 1. Therefore, p only depends on eigenvectors!Ramis Movassagh arXiv: 1012.5039; Phys. Rev. Lett. 107, 097205 (2011)
Gap of IgnorancePhilosophy of Isotropic Entanglement
Theory of Isotropic EntanglementThe Method of Isotropic Entanglement
More Numerical results
Local terms: Wishart matrices
0 10 20 30 40 50 60 70 80 90 100 1100
0.005
0.01
0.015
0.02
0.025
0.03N=11, d=2, r=4, trials=2800
Eigenvalue
Density
p = 0
p = 1
Exact Diagonalization
I.E. Theory: p = 0.73
Ramis Movassagh arXiv: 1012.5039; Phys. Rev. Lett. 107, 097205 (2011)
Gap of IgnorancePhilosophy of Isotropic Entanglement
Theory of Isotropic EntanglementThe Method of Isotropic Entanglement
More Numerical results
Suppose you have the first four moments
−3 −2 −1 0 1 2 30
0.5
1
1.5
2
2.54−moment matching methods vs. IE: binomial local terms
Eigenvalue
Density
Exact Diagonalization
I.E. Theory: p = 0.046
Pearson
Gram−Charlier
Ramis Movassagh arXiv: 1012.5039; Phys. Rev. Lett. 107, 097205 (2011)
Gap of IgnorancePhilosophy of Isotropic Entanglement
Theory of Isotropic EntanglementThe Method of Isotropic Entanglement
More Numerical results
Suppose you have the first four moments
0 10 20 30 40 50 60 700
0.01
0.02
0.03
0.04
0.054−moment matching methods vs. IE: Wishart local terms
Eigenvalue
Density
Exact Diagonalization
I.E. Theory: p = 0.42817
Pearson
Gram−Charlier
Ramis Movassagh arXiv: 1012.5039; Phys. Rev. Lett. 107, 097205 (2011)
Gap of IgnorancePhilosophy of Isotropic Entanglement
Theory of Isotropic EntanglementThe Method of Isotropic Entanglement
More Numerical results
Summary: Method of Isotropic Entanglement
} }+ +
A B
}+
add random variables ≡Classical convolution of densities
add random variables ≡Classical convolution of densities
HH 1,21,2 HH N-1,NN-1,N HH 2,32,3HH N-2,N-1N-2,N-1
Several options for adding random variables
Isotropic (or free) convolution ≡ dνIso = dν
A ⊞Isodν
B
Classical convolution ≡dνC = dν
A ∗ dν
B
pIsotropicp = 0
Classicalp = 1
Isotropically Entangled ≡ (1-p) dνIso
+ p dνC
Quantum convolution ≡ dνq = dν
A ⊞qdν
B ≈
Isotropically Entangled
Ramis Movassagh arXiv: 1012.5039; Phys. Rev. Lett. 107, 097205 (2011)
Gap of IgnorancePhilosophy of Isotropic Entanglement
Theory of Isotropic EntanglementThe Method of Isotropic Entanglement
More Numerical results
Others got excited about it too!
Error analysis of free probability approximations to the density of states of disordered systems
Jiahao Chen,⇤ Eric Hontz,† Jeremy Moix,‡ Matthew Welborn,§ and Troy Van Voorhis¶
Department of Chemistry, Massachusetts Institute of Technology,77 Massachusetts Avenue, Cambridge, Massachusetts 02139, USA
Alberto Suarez⇤⇤Departamento de Ingenierıa Informatica, Escuela Politecnica Superior,
Universidad Autonoma de Madrid, Ciudad Universitaria de Cantoblanco,Calle Francisco Tomas y Valiente, 11 E-28049 Madrid, Spain
Ramis Movassagh†† and Alan Edelman‡‡
Department of Mathematics, Massachusetts Institute of Technology,77 Massachusetts Avenue, Cambridge, Massachusetts 02139, USA
Theoretical studies of localization, anomalous diffusion and ergodicity breaking require solvingthe electronic structure of disordered systems. We use free probability to approximate the ensemble-averaged density of states without exact diagonalization. We present an error analysis that quantifiesthe accuracy using a generalized moment expansion, allowing us to distinguish between differentapproximations. We identify an approximation that is accurate to the eighth moment across all noisestrengths, and contrast this with the perturbation theory and isotropic entanglement theory.
PACS numbers: 71.23.An, 71.23.-k
Disordered materials have long been of interest for theirunique physics such as localization [1, 2], anomalous dif-fusion [3, 4] and ergodicity breaking [5]. Their propertieshave been exploited for applications as diverse as quantumdots [6, 7], magnetic nanostructures [8], disordered met-als [9, 10], and bulk heterojunction photovoltaics [11–13].Despite this, theoretical studies are complicated by the needto calculate the electronic structure of the respective systemsin the presence of random atomic nuclear positions. Con-ventional electronic structure theories can only be used inconjunction with explicit sampling of thermodynamicallyaccessible regions of phase space, which make such calcu-lations enormously more expensive than usual single-pointcalculations [14].
Alternatively, ensemble-averaged quantities may be com-puted or approximated using random matrix theory. In par-ticular, techniques from free probability theory allow thecomputation of eigenvalues of sums of certain matrices with-out rediagonalizing the matrix sums [15]. While this hasbeen proposed as a tool applicable to general random matri-ces [16] and has been used for similar purposes in quantumchromodynamics [17], we are not aware of any quantificationof the accuracy of this approximation in practice. We provideherein a general framework for quantitatively estimating theerror in such situations. We find that this allows us to under-stand the relative performances of various approximations,and furthermore characterize the degree of accuracy system-atically in terms of discrepancies in particular moments ofthe probability distribution functions (PDFs).
Quantifying the error in approximating a PDF using freeprobability.— We propose to quantify the deviation betweentwo PDFs using moment expansions. Such expansions arewidely used to describe corrections to the central limit the-orem and deviations from normality, and are often appliedin the form of Gram–Charlier and Edgeworth series [18, 19].Similarly, deviations from non-Gaussian reference PDFs canbe quantified using generalized moment expansions. For
two PDFs w (x) and w (x) with finite cumulants k1, k2, . . .and k1, k2, . . . , and moments µ1, µ2, . . . and µ1, µ2, . . . respec-tively, we can define a formal differential operator whichtransforms w into w and is given by [18, 20]
w (x) = exp
"•
Ân=1
kn � kn
n!
✓� d
dx
◆n#
w (x) . (1)
This operator is parameterized completely by the cumulantsof both distributions.
The first k for which the cumulants kk and kk differ thenallows us to define a degree to which the approximationw ⇡ w is valid. Expanding the exponential and using thewell-known relationship between cumulants and momentsallows us to state that if the first k � 1 cumulants agree, butthe kth cumulants differ, that this is equivalent to specifyingthat
w (x) = w (x) +µk � µk
k!(�1)k w(k) (x) + O
⇣w(k+1)
⌘. (2)
At this point we make no claim on the convergence ofthe series defined by the expansion of (1), but use it as ajustification for calculating the error term defined in (2). Wewill examine this claim later.
The free convolution.— We now take the PDFs to be DOSsof random matrices. For a random matrix Z, the DOS isdefined in terms of the eigenvalues
nl
(m)n
oof the M samples
Z1, . . . , Zm, . . . , ZM according to
r(Z) (x) = limM!•
1M
M
Âm=1
1N
N
Ân=1
d⇣
x � l(m)n
⌘. (3)
The central idea using free probability to calculate approxi-mate DOSs is to split the Hamiltonian H = A + B into twomatrices A and B whose DOSs, r(A) and r(B) respectively,can be determined easily. The eigenvalues of the sum isin general not the sum of the eigenvalues; instead, we ap-proximate the exact DOS with the free convolution A � B,
arX
iv:1
202.
5839
v1 [
cond
-mat
.dis
-nn]
27
Feb
2012
Phys. Rev. Lett. 109, 036403 (2012)
Ramis Movassagh arXiv: 1012.5039; Phys. Rev. Lett. 107, 097205 (2011)
Gap of IgnorancePhilosophy of Isotropic Entanglement
Theory of Isotropic EntanglementThe Method of Isotropic Entanglement
More Numerical results
Question for you: Why do we do so well?
m5=1mETr
(A5+5A4QT
• BQ•+5A3QT• B2Q•+5A2QT
• B3Q•+5A(AQT• BQ•)
2+
5(AQT• BQ•)
2QT• BQ•+5AQT
• B4Q•+B5)
(1)
Ramis Movassagh arXiv: 1012.5039; Phys. Rev. Lett. 107, 097205 (2011)
Gap of IgnorancePhilosophy of Isotropic Entanglement
Theory of Isotropic EntanglementThe Method of Isotropic Entanglement
More Numerical results
Question for you: Why do we do so well?
ETr{. . .Q−1B≥1QA≥1Q−1B≥1Q . . .
}≤
ETr{. . .Q−1
q B≥1QqA≥1Q−1q B≥1Qq . . .
}≤
ETr{. . .B≥1A≥1B≥1 . . .
}.
e.g. ETr{(
AQ−1BQ)k}≤ ETr
{(AQ−1
q BQq)k}≤ ETr
{(AB)k
}
K > 2
Ramis Movassagh arXiv: 1012.5039; Phys. Rev. Lett. 107, 097205 (2011)